When I overlay a Surfer post map with 3D label lines on a surface map, the 3D label lines disappear.

Overlays on a surface map are combined into a single image "texture" that is applied to the surface, so 3D label lines are not supported with surface map overlays. To add your vote to this request on our suggestion file, please email surfersupport@goldensoftware.com.

You may be able to use a 3D wireframe instead of a 3D surface. The 3D wireframe map does support 3D label lines for overlaid post maps. Alternatively, you can overlay the post map on a wireframe map, turn off the wireframe visibility, and align it with the surface map to achieve a similar effect.